How do you rig eyeballs without affecting the rest of the face?
Here's my workflow for eyes:
First clean any weight paint of your eye bone from your face mesh.
If I understood it correctly face mesh is connected to the eyes? If that's the case then select face in object mode > Edit mode > select eye faces > ctrl + I > H, this will hide everything except eye(s).

Select bone of an eye in pose mode > Ctrl + tab.
Now we have to apply our selection from edit mode.



Click on the second from the right side icon.

Weight paint whole eye with value 1(red). This creates vertex group with weights for my eye bone.

Hope this helps.